IMO 1997

Problem 1. In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares.

Let $S_1$ be the total area of the black part of the triangle and $S_2$ be the total area of the white part. Let

$f(m,n)=|S_1-S_2|.$

• Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

• Prove that $f(m,n)\le \frac{1}{2}\max \{m,n\}$ for all $m$ and $n$.

• Show that there is no constant $C$ such that $f(m,n)<C$ for all $m$ and $n$.

Problem 2. The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that

$AU=TB+TC.$

Problem 3. Let $x_1,x_2,\dots ,x_n$ be real numbers satisfying the conditions

$\left|x_1+x_2+\cdots +x_n\right|=1$

and

$\left|x_i\right|\le \frac{n+1}{2}i=1,2,\dots ,n.$

Show that there exists a permutation $y_1,y_2,\dots ,y_n$ of $x_1,x_2,\dots ,x_n$ such that

$\left|y_1+2y_2+\cdots +n{y}_n\right|\le \frac{n+1}{2}.$

Problem 4. An $n\times n$ matrix whose entries come from the set $S=\{1,2,\dots ,2n-1\}$ is called a silver matrix if, for each $i=1,2,\dots ,n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

• there is no silver matrix for $n=1997$;

• silver matrices exist for infinitely many values of $n$.

Problem 5. Find all pairs $(a,b)$ of integers $a,b\ge 1$ that satisfy the equation

$a^{b^2}=b^a.$

Problem 6. For each positive integer $n$ , let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways:

$4;2+2;2+1+1;1+1+1+1.$

Prove that, for any integer $n\ge 3$,

$2^{n^2/4}<f(2^n)<2^{n^2/2}.$